Is there such a thing as "coaxioms"?

Question

Loosely speaking, category theory suggests that "equivalence relation" is dual to "subset." Anyway, since axioms correspond to subclasses of the possible models of a signature, is there some notion of "coaxiom" that corresponds to making "equivalent" (that is, isomorphic) previously non-equivalent models?

For example, it would be cool if by adding coaxioms to the notion of a metric space, we could obtain the notion of "metric space up to uniform scaling of distances." Similarly, it would be cool if by adding coaxioms to the notion of a topological space, we could obtain the notion of "topological space up to homotopy equivalence."

Answer 1

Not really what you are looking for, but I wanted to provide one type of answer.

Typically, axioms are represented equationally (this is, for instance, the means used in universal algebra and the categorial methods inspired by it). The models form a variety over the structure space.

In a real sense, the examples given in the question are equational, and are perfectly capable of axiomatic representation in the natural way.

The natural view of coaxioms may be in coequations. There is a good amount of literature on coequational logics. Although not quite what you are looking for, the semantic purpose of coequational formulations does play some interesting roles in computer science.

Also, there is the idea of covarieties that plays a dual role to models of a theory. This follows the same equational representation of axiom line of thought.